Tell whether the function represents exponential growth or exponential decay. Identify the rate of increase or decrease and the initial amount. Then find the value of \( A \) when \( t=5 \). 21. \( A(t)=1500(0.96)^{t} \)

To determine whether the function represents exponential growth or exponential decay, we need to analyze the coefficient of the exponential term. Given function: \( A(t) = 1500(0.96)^t \) In this function, the coefficient of the exponential term is 0.96, which is less than 1. This indicates that the function represents exponential decay. The rate of decrease is 0.96, which means that the amount decreases by 4% each time the variable \( t \) increases by 1. The initial amount can be found by setting \( t = 0 \): \( A(0) = 1500(0.96)^0 = 1500 \) Therefore, the initial amount is 1500. To find the value of \( A \) when \( t = 5 \), we substitute \( t = 5 \) into the function: \( A(5) = 1500(0.96)^5 \) Now, let's calculate the value of \( A(5) \). Calculate the value by following steps: - step0: Calculate: \(1500\times 0.96^{5}\) - step1: Convert the expressions: \(1500\left(\frac{24}{25}\right)^{5}\) - step2: Simplify: \(1500\times \frac{24^{5}}{25^{5}}\) - step3: Rewrite the expression: \(125\times 12\times \frac{24^{5}}{25^{5}}\) - step4: Rewrite the expression: \(5^{3}\times 12\times \frac{24^{5}}{5^{10}}\) - step5: Reduce the numbers: \(12\times \frac{24^{5}}{5^{7}}\) - step6: Multiply: \(\frac{12\times 24^{5}}{5^{7}}\) The value of \( A \) when \( t = 5 \) is approximately 1223.0590464.